Solve for R1 (2R_1 square root of R_1)/(R_1+2 square root of R_1)=15

$\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} = 15$

Multiply both sides by $R_{1} + 2 \sqrt{R_{1}}$ .

$\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} (R_{1} + 2 \sqrt{R_{1}}) = 15 (R_{1} + 2 \sqrt{R_{1}})$

Simplify.

Simplify the left side.

Cancel the common factor of $R_{1} + 2 \sqrt{R_{1}}$ .

Cancel the common factor.

$\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} (R_{1} + 2 \sqrt{R_{1}}) = 15 (R_{1} + 2 \sqrt{R_{1}})$

Rewrite the expression.

$2 R_{1} \sqrt{R_{1}} = 15 (R_{1} + 2 \sqrt{R_{1}})$

Simplify the right side.

Simplify $15 (R_{1} + 2 \sqrt{R_{1}})$ .

Apply the distributive property.

$2 R_{1} \sqrt{R_{1}} = 15 R_{1} + 15 (2 \sqrt{R_{1}})$

Multiply $2$ by $15$ .

$2 R_{1} \sqrt{R_{1}} = 15 R_{1} + 30 \sqrt{R_{1}}$

Solve for $R_{1}$ .

Solve for $\sqrt{R_{1}}$ .

Subtract $30 \sqrt{R_{1}}$ from both sides of the equation.

$2 R_{1} \sqrt{R_{1}} - 30 \sqrt{R_{1}} = 15 R_{1}$

Factor $2 \sqrt{R_{1}}$ out of $2 R_{1} \sqrt{R_{1}} - 30 \sqrt{R_{1}}$ .

Factor $2 \sqrt{R_{1}}$ out of $2 R_{1} \sqrt{R_{1}}$ .

$2 \sqrt{R_{1}} R_{1} - 30 \sqrt{R_{1}} = 15 R_{1}$

Factor $2 \sqrt{R_{1}}$ out of $- 30 \sqrt{R_{1}}$ .

$2 \sqrt{R_{1}} R_{1} + 2 \sqrt{R_{1}} \cdot - 15 = 15 R_{1}$

Factor $2 \sqrt{R_{1}}$ out of $2 \sqrt{R_{1}} R_{1} + 2 \sqrt{R_{1}} \cdot - 15$ .

$2 \sqrt{R_{1}} (R_{1} - 15) = 15 R_{1}$

Divide each term in $2 \sqrt{R_{1}} (R_{1} - 15) = 15 R_{1}$ by $2 (R_{1} - 15)$ and simplify.

Divide each term in $2 \sqrt{R_{1}} (R_{1} - 15) = 15 R_{1}$ by $2 (R_{1} - 15)$ .

$\frac{2 \sqrt{R_{1}} (R_{1} - 15)}{2 (R_{1} - 15)} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{R_{1}} (R_{1} - 15)}{2 (R_{1} - 15)} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Rewrite the expression.

$\frac{\sqrt{R_{1}} (R_{1} - 15)}{R_{1} - 15} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Cancel the common factor of $R_{1} - 15$ .

Cancel the common factor.

$\frac{\sqrt{R_{1}} (R_{1} - 15)}{R_{1} - 15} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Divide $\sqrt{R_{1}}$ by $1$ .

$\sqrt{R_{1}} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{R_{1}}}^{2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{R_{1}}$ as ${R_{1}}^{\frac{1}{2}}$ .

${({R_{1}}^{\frac{1}{2}})}^{2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Simplify the left side.

Simplify ${({R_{1}}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({R_{1}}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${R_{1}}^{\frac{1}{2} \cdot 2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${R_{1}}^{\frac{1}{2} \cdot 2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Rewrite the expression.

${R_{1}}^{1} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Simplify.

$R_{1} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Simplify the right side.

Simplify ${(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$ .

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{15 R_{1}}{2 (R_{1} - 15)}$ .

$R_{1} = \frac{{(15 R_{1})}^{2}}{{(2 (R_{1} - 15))}^{2}}$

Apply the product rule to $15 R_{1}$ .

$R_{1} = \frac{15^{2} {R_{1}}^{2}}{{(2 (R_{1} - 15))}^{2}}$

Apply the product rule to $2 (R_{1} - 15)$ .

$R_{1} = \frac{15^{2} {R_{1}}^{2}}{2^{2} {(R_{1} - 15)}^{2}}$

Raise $15$ to the power of $2$ .

$R_{1} = \frac{225 {R_{1}}^{2}}{2^{2} {(R_{1} - 15)}^{2}}$

Raise $2$ to the power of $2$ .

$R_{1} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}}$

Solve for $R_{1}$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 4 {(R_{1} - 15)}^{2}$

The LCM of one and any expression is the expression.

$4 {(R_{1} - 15)}^{2}$

Multiply each term in $R_{1} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}}$ by $4 {(R_{1} - 15)}^{2}$ to eliminate the fractions.

Multiply each term in $R_{1} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}}$ by $4 {(R_{1} - 15)}^{2}$ .

$R_{1} (4 {(R_{1} - 15)}^{2}) = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} (4 {(R_{1} - 15)}^{2})$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$4 R_{1} {(R_{1} - 15)}^{2} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} (4 {(R_{1} - 15)}^{2})$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$4 R_{1} {(R_{1} - 15)}^{2} = 4 \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Cancel the common factor of $4$ .

Factor $4$ out of $4 {(R_{1} - 15)}^{2}$ .

$4 R_{1} {(R_{1} - 15)}^{2} = 4 \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Cancel the common factor.

$4 R_{1} {(R_{1} - 15)}^{2} = 4 \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Rewrite the expression.

$4 R_{1} {(R_{1} - 15)}^{2} = \frac{225 {R_{1}}^{2}}{{(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Cancel the common factor of ${(R_{1} - 15)}^{2}$ .

Cancel the common factor.

$4 R_{1} {(R_{1} - 15)}^{2} = \frac{225 {R_{1}}^{2}}{{(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Rewrite the expression.

$4 R_{1} {(R_{1} - 15)}^{2} = 225 {R_{1}}^{2}$

Solve the equation.

Move all terms containing $R_{1}$ to the left side of the equation.

Subtract $225 {R_{1}}^{2}$ from both sides of the equation.

$4 R_{1} {(R_{1} - 15)}^{2} - 225 {R_{1}}^{2} = 0$

Simplify each term.

Rewrite ${(R_{1} - 15)}^{2}$ as $(R_{1} - 15) (R_{1} - 15)$ .

$4 R_{1} ((R_{1} - 15) (R_{1} - 15)) - 225 {R_{1}}^{2} = 0$

Expand $(R_{1} - 15) (R_{1} - 15)$ using the FOIL Method.

Apply the distributive property.

$4 R_{1} (R_{1} (R_{1} - 15) - 15 (R_{1} - 15)) - 225 {R_{1}}^{2} = 0$

Apply the distributive property.

$4 R_{1} (R_{1} R_{1} + R_{1} \cdot - 15 - 15 (R_{1} - 15)) - 225 {R_{1}}^{2} = 0$

Apply the distributive property.

$4 R_{1} (R_{1} R_{1} + R_{1} \cdot - 15 - 15 R_{1} - 15 \cdot - 15) - 225 {R_{1}}^{2} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $R_{1}$ by $R_{1}$ .

$4 R_{1} ({R_{1}}^{2} + R_{1} \cdot - 15 - 15 R_{1} - 15 \cdot - 15) - 225 {R_{1}}^{2} = 0$

Move $- 15$ to the left of $R_{1}$ .

$4 R_{1} ({R_{1}}^{2} - 15 \cdot R_{1} - 15 R_{1} - 15 \cdot - 15) - 225 {R_{1}}^{2} = 0$

Multiply $- 15$ by $- 15$ .

$4 R_{1} ({R_{1}}^{2} - 15 R_{1} - 15 R_{1} + 225) - 225 {R_{1}}^{2} = 0$

Subtract $15 R_{1}$ from $- 15 R_{1}$ .

$4 R_{1} ({R_{1}}^{2} - 30 R_{1} + 225) - 225 {R_{1}}^{2} = 0$

Apply the distributive property.

$4 R_{1} {R_{1}}^{2} + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Simplify.

Multiply $R_{1}$ by ${R_{1}}^{2}$ by adding the exponents.

Move ${R_{1}}^{2}$ .

$4 ({R_{1}}^{2} R_{1}) + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Multiply ${R_{1}}^{2}$ by $R_{1}$ .

Raise $R_{1}$ to the power of $1$ .

$4 ({R_{1}}^{2} {R_{1}}^{1}) + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 {R_{1}}^{2 + 1} + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Add $2$ and $1$ .

$4 {R_{1}}^{3} + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Rewrite using the commutative property of multiplication.

$4 {R_{1}}^{3} + 4 \cdot - 30 R_{1} R_{1} + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Multiply $225$ by $4$ .

$4 {R_{1}}^{3} + 4 \cdot - 30 R_{1} R_{1} + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Simplify each term.

Multiply $R_{1}$ by $R_{1}$ by adding the exponents.

Move $R_{1}$ .

$4 {R_{1}}^{3} + 4 \cdot - 30 (R_{1} R_{1}) + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Multiply $R_{1}$ by $R_{1}$ .

$4 {R_{1}}^{3} + 4 \cdot - 30 {R_{1}}^{2} + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Multiply $4$ by $- 30$ .

$4 {R_{1}}^{3} - 120 {R_{1}}^{2} + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Subtract $225 {R_{1}}^{2}$ from $- 120 {R_{1}}^{2}$ .

$4 {R_{1}}^{3} - 345 {R_{1}}^{2} + 900 R_{1} = 0$

Factor $R_{1}$ out of $4 {R_{1}}^{3} - 345 {R_{1}}^{2} + 900 R_{1}$ .

Factor $R_{1}$ out of $4 {R_{1}}^{3}$ .

$R_{1} (4 {R_{1}}^{2}) - 345 {R_{1}}^{2} + 900 R_{1} = 0$

Factor $R_{1}$ out of $- 345 {R_{1}}^{2}$ .

$R_{1} (4 {R_{1}}^{2}) + R_{1} (- 345 R_{1}) + 900 R_{1} = 0$

Factor $R_{1}$ out of $900 R_{1}$ .

$R_{1} (4 {R_{1}}^{2}) + R_{1} (- 345 R_{1}) + R_{1} \cdot 900 = 0$

Factor $R_{1}$ out of $R_{1} (4 {R_{1}}^{2}) + R_{1} (- 345 R_{1})$ .

$R_{1} (4 {R_{1}}^{2} - 345 R_{1}) + R_{1} \cdot 900 = 0$

Factor $R_{1}$ out of $R_{1} (4 {R_{1}}^{2} - 345 R_{1}) + R_{1} \cdot 900$ .

$R_{1} (4 {R_{1}}^{2} - 345 R_{1} + 900) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$R_{1} = 0$

$4 {R_{1}}^{2} - 345 R_{1} + 900 = 0$

Set $R_{1}$ equal to $0$ .

$R_{1} = 0$

Set $4 {R_{1}}^{2} - 345 R_{1} + 900$ equal to $0$ and solve for $R_{1}$ .

Set $4 {R_{1}}^{2} - 345 R_{1} + 900$ equal to $0$ .

$4 {R_{1}}^{2} - 345 R_{1} + 900 = 0$

Solve $4 {R_{1}}^{2} - 345 R_{1} + 900 = 0$ for $R_{1}$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 4$ , $b = - 345$ , and $c = 900$ into the quadratic formula and solve for $R_{1}$ .

$\frac{345 \pm \sqrt{{(- 345)}^{2} - 4 \cdot (4 \cdot 900)}}{2 \cdot 4}$

Simplify.

Simplify the numerator.

Raise $- 345$ to the power of $2$ .

$R_{1} = \frac{345 \pm \sqrt{119025 - 4 \cdot 4 \cdot 900}}{2 \cdot 4}$

Multiply $- 4$ by $4$ .

$R_{1} = \frac{345 \pm \sqrt{119025 - 16 \cdot 900}}{2 \cdot 4}$

Multiply $- 16$ by $900$ .

$R_{1} = \frac{345 \pm \sqrt{119025 - 14400}}{2 \cdot 4}$

Subtract $14400$ from $119025$ .

$R_{1} = \frac{345 \pm \sqrt{104625}}{2 \cdot 4}$

Rewrite $104625$ as $15^{2} \cdot 465$ .

Factor $225$ out of $104625$ .

$R_{1} = \frac{345 \pm \sqrt{225 (465)}}{2 \cdot 4}$

Rewrite $225$ as $15^{2}$ .

$R_{1} = \frac{345 \pm \sqrt{15^{2} \cdot 465}}{2 \cdot 4}$

Pull terms out from under the radical.

$R_{1} = \frac{345 \pm 15 \sqrt{465}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$R_{1} = \frac{345 \pm 15 \sqrt{465}}{8}$

The final answer is the combination of both solutions.

$R_{1} = \frac{345 + 15 \sqrt{465}}{8}, \frac{345 - 15 \sqrt{465}}{8}$

The final solution is all the values that make $R_{1} (4 {R_{1}}^{2} - 345 R_{1} + 900) = 0$ true.

$R_{1} = 0, \frac{345 + 15 \sqrt{465}}{8}, \frac{345 - 15 \sqrt{465}}{8}$

Exclude the solutions that do not make $\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} = 15$ true.

$R_{1} = \frac{345 + 15 \sqrt{465}}{8}$

The result can be shown in multiple forms.

Exact Form:

$R_{1} = \frac{345 + 15 \sqrt{465}}{8}$

Decimal Form:

$R_{1} = 83.55723497 \dots$

Solve for R1 (2R_1 square root of R_1)/(R_1+2 square root of R_1)=15

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